Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year


IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

Online
ISSN
2191-0294
See all formats and pricing
More options …
Volume 18, Issue 7-8

Issues

A Fractional q-difference Equation with Integral Boundary Conditions and Comparison Theorem

Jing Ren / Chengbo Zhai
Published Online: 2017-10-28 | DOI: https://doi.org/10.1515/ijnsns-2017-0056

Abstract

In this article, we mainly prove the existence of extremal solutions for a fractional q-difference equation involving Riemann–Lioville type fractional derivative with integral boundary conditions. A comparison theorem under weak conditions is also build, and then we apply the comparison theorem, monotone iterative technique and lower–upper solution method to prove the existence of extremal solutions. Moreover, we can construct two iterative schemes approximating the extremal solutions of the fractional q-difference equation with integral boundary conditions. In the last section, a simple example is presented to illustrate the main result.

Keywords: fractional q-difference equation; comparison theorem; integral boundary conditions; extremal solutions; iterative method

MSC 2010: 34B18; 33D05

References

  • [1]

    Li X., Han Z., Sun S. and Zhao P., Existence of solutions for fractional q-difference equation with mixed nonlinear boundary conditions, Adv. Differ. Equ. 326 (2014), 1–11.Web of ScienceGoogle Scholar

  • [2]

    Li X., Han Z. and Li X., Boundary value problems of fractional q-difference Schróinger equations, Appl. Math. Lett. 46 (2015), 100–105.CrossrefGoogle Scholar

  • [3]

    Ma J., Yang J., Existence of solutions for multi-point boundary value problem of fractional q-difference equation, Electron. J. Qual. Theory Differ. Equ. 92 (2011), 1–10.Google Scholar

  • [4]

    Almeida R. and Martins N., Existence results for fractional q-difference equations of order α Η]2, 3[ with three-point boundary conditions, Comm. Nonl. Sci. Nume. Simu. 19 (2014), 1675–1685.CrossrefGoogle Scholar

  • [5]

    Li X., Han Z. and Sun S., Existence of positive solutions of nonlinear fractional q-difference equation with parameter, Adv. Differ. Equ. 260 (2013), 1–13.Web of ScienceGoogle Scholar

  • [6]

    Ahmad B., Etemad S., Ettefagh M. and Rezapour S., On the existence of solutions for fractional q-difference inclusions with q-antiperiodic boundary conditions, Bull. Math. Soc. Sci. Math. Roumanie. 59 (2016), 119–134.Google Scholar

  • [7]

    Ahmad B., Ntouyas S. K. and Purnaras I. K., Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations, Adv. Differ. Equ. 140 (2012), 1–15.Web of ScienceGoogle Scholar

  • [8]

    Liang S. and Zhang J., Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences, J. Appl. Math. Comput. 40 (2012), 277–288.CrossrefGoogle Scholar

  • [9]

    Miao F. and Liang S., Uniqueness of positive solutions for fractional q-difference boundary-value problems with p-Laplacian operator, Electron J. Differ. Equ. 174 (2013), 1–11.Google Scholar

  • [10]

    Li Y. and Yang W., Monotone iterative method for nonlinear fractional q-difference equations with integral boundary conditions, Adv. Differ. Equ. 294 (2015), 1–10.Web of ScienceGoogle Scholar

  • [11]

    Khodabakhshi N. and Vaezpour S. M., Existence and uniqueness of positive solution for a class of boundary value problems with fractional q-differences, J. Nonl. Conv. Anal. 16 (2015), 375–384.Google Scholar

  • [12]

    Wang G., Sudsutad W., Zhang L. H. and Tariboon J., Monotone iterative technique for a nonlinear fractional q-difference equation of Caputo type, Adv. Differ. Equ. 211 (2016), 1–11.Web of ScienceGoogle Scholar

  • [13]

    Graef J. R. and Kong L., Existence of positive solutions to a higher order singular boundary value problem with fractional q-derivatives, Fract. Calc. Appl. Anal. 16 (2013), 695–708.Web of ScienceGoogle Scholar

  • [14]

    Ferreira R. A. C., Nontrivial solutions for fractional q-difference boundary value problems, Electron J. Qual. Theory. Differ. Equ. 70 (2010), 1–10.Google Scholar

  • [15]

    Wang X., Wang L. and Zeng Q., Fractional differential equations with integral boundary conditions, J. Nonl. Sci. Appl. 8 (2015), 309–314.Google Scholar

  • [16]

    Zhou W. and Liu H., Existence solutions for boundary value problem of nonlinear fractional q-difference equations, Adv. Differ. Equ. 113 (2013), 1–12.Web of ScienceGoogle Scholar

  • [17]

    Yang W., Positive solution for fractional q-difference boundary value problems with Φ-Laplacian operator, Bulle. Malays. Math. Soci. 36 (2013), 1195–1203.Google Scholar

  • [18]

    Zhao Y., Chen H. and Zhang Q., Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions, Adv. Differ. Equ. 48 (2013), 1–15.Web of ScienceGoogle Scholar

  • [19]

    Agarwal R. P., Wang G., Ahmad B., Zhang L., Hobiny A. and Monaquel S., On existence of solutions for nonlinear q-difference equations with nonlocal q-integral boundary conditions, Math. Model. Anal. 20 (5) (2015), 604–618.Google Scholar

  • [20]

    Jiang M. and Zhong S., Existence of extremal solutions for a nonlinear fractional q-difference system, Mediterr. J. Math. 13 (1) (2016), 279–299.CrossrefWeb of ScienceGoogle Scholar

  • [21]

    Wang J., Fečkan M. and Zhou Y., A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), 806–831.Google Scholar

  • [22]

    Wang J., Fečkan M. and Zhou Y., Center stable manifold for planar fractional damped equations, Appl. Math. Comput. 296 (2017), 257–269.Web of ScienceGoogle Scholar

  • [23]

    Li M. and Wang J., Finite time stability of fractional delay differential equations, Appl. Math. Lett. 64 (2017), 170–176.CrossrefWeb of ScienceGoogle Scholar

  • [24]

    Zhai C. and Ren J., Positive and negative solutions of a boundary value problem for a fractional q-difference equation, Adv. Diff. Equ. 82 (2017), 1–13.Google Scholar

  • [25]

    Rajkovič P. M., Marinkovič S. D. and Stankovič M. S., Fractional integrals and derivatives in q-calculus, Appl. Anal. Disc. Math. 1 (2007), 1–13.Google Scholar

  • [26]

    Al-Salam W. A., Some fractional q-integrals and q-derivatives, Proc. Edinb. Math. Soc. 15 (1966–1967), 135–140.CrossrefGoogle Scholar

  • [27]

    Annaby M. H. and Mansour Z. S., q-Fractional calculus and equations, Lecture notes in mathematics. Vol. 2012.Web of ScienceGoogle Scholar

About the article

Published Online: 2017-10-28

Published in Print: 2017-12-20


This paper was supported financially by the Youth Science Foundation of China (11201272), Shanxi Province Science Foundation (2015011005) and 131 Talents Project of Shanxi Province (2015).


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 18, Issue 7-8, Pages 575–583, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0056.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in